Some remarks on quasi-Hermitian operators

TL;DR

This paper investigates the structure and properties of quasi-Hermitian operators, especially unbounded ones, and explores their applications in pseudo-Hermitian quantum mechanics, extending the concept of similarity between operators.

Contribution

It introduces generalizations of operator similarity and systematically analyzes various types of quasi-Hermitian operators, including unbounded cases.

Findings

Analysis of unbounded metric operators in Hilbert spaces

Generalizations of similarity between operators

Application insights into pseudo-Hermitian quantum mechanics

Abstract

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally we discuss their application in the so-called pseudo-Hermitian quantum mechanics.